As a very general assessment, I don't think reading this book will make you walk away a communist, but I sure hope it would make you walk away with very serious reservations about capitalism. Eagleton's defenses of Marx and his critiques of capitalism are astute, but I would say that he does a better job of showing why we should ditch capitalism than he does showing why we should adopt Marxism. This imbalance is partly a consequence of the structure, but it remains the case that he doesn't explain why Marxism is better than the other non-capitalist options, because there are other options. For instance, there are more kinds of communism than Marxism, but there are also things that aren't capitalism that also aren't communism. Marx wasn't the only one who has provided resources for anti-capitalist thought, cf the Gospel of Luke.
I haven't much more to say about Why Marx Was Right as a whole; when your defense consists mainly of refutations of refutations, you don't work out much of a larger system for me to interact with (not that he has to, since he's working with a preexisting system, i.e. Marxism). However, Eagleton's greatest strength is probably his habit of incorporating lesser-known parts of Marxist thought into his defenses. Not only are these claims probable-sounding and interesting, but if they turn out to be true they'll be rather important. I'd like to share and discuss some of these ideas here.
1. There is such a thing as a universal human nature. In particular, humans have material needs, and this fundamentally effects our behaviour. This feature doesn't differentiate us from other animals. What does differentiate us from other animals is our ability to self-actualize; our ability to assess, learn about, and improve ourselves is also part of our nature.
I tend to be pretty hard on the idea of a universal human nature (in fact, if there's one argument that will tempt me to start ignoring you entirely, it's an appeal to human nature), so it's generally good to be reminded that there is at least a bare minimum of such universal nature and that that minimum determines quite a lot about us.
2. Thought cannot really be abstract; it is always grounded in materiality. First, physical bodies are the condition of thought in the first place; you can't have minds without brains. Second, thought originates from its ability to fulfill material needs. If thought did not fulfill material needs, we would never have evolved or practiced it. All thought, abstract or otherwise, has physical needs beneath it.
Marx would go on to say that these physical needs to some extent determine all thought, regardless of how apparently abstract/disinterested, or that we should at least not be surprised to find that our abstract thought covertly (even to us) serves our material needs. I find this logical process a little far-fetched--the origins of a thing do not so totally control what we can use it for--but I feel like the case is overstated rather than incorrect, especially considering what we know about human cognitive biases and how bad highly abstract thinkers tend to be at introspection/self-awareness.
3. Following #1 and #2, Marx was somewhat anti-philosophical. Eagleton says that Marx has an aversion to abstract thought. Just as thought is based in materiality, any system of thought that deserves our attention must begin with the recognition that human beings have material needs and emotions bound up with those needs. There cannot be reason divorced from those needs or emotions. Further, any philosophy that is not about those needs is omitting basic data about the world. (Of course, Marxists can be just as guilty of this, in that many of them tend to think of people as non-gendered, etc.)
We might be able to imagine sci-fi contexts in which there are perfectly rational creatures, but it wouldn't tell us much that would be of any use to us. Further, refering to the next point, I don't think we can accurately imagine a disinterested or immaterial intelligence anyway.
4. Any radical break with the present--as in communism, but you can think of a lot of things, like abolition or gender equity or the Protestant Reformation--will result in a society that cannot be predicted from the present. In other words, even Marx doesn't really claim to know what a communist society would look like or how it would be organized. That sort of thing could only be figured out in situ, without the pressures on thought, philosophy, etc. that our current economic system makes. But, more importantly, a radical break from the present, at least in the case of Marxist revolution, would mean that a whole different set of people will be making the important decisions, and those decisions wouldn't really be freely theirs if they could be predicted in advance. This might make communism hard to sell because people want to know what they're buying, but it is unavoidably true.
What interests me about this idea isn't just that it explains the lack of specifics a lot of communists give about life under communism (excepting Fourier, of course), but that it seems relevant to a lot of arguments I see people getting into. Whenever someone makes claims about what would happen if some radical change took place, I want to shout that they can't possibly know that; if we had any evidence to go on, then it would not be a radical change because it would have happened already. It might seem like a decidedly pessimistic view of the imagination, but it's really just a pessimistic view of our imagination's predictive power.
5. Since I'm rather interested in praxis as a word (I tend to be more worried about orthopraxy than orthodoxy, but I also just like Greek words with ties to theology), I was interested to learn that Marx talks about praxis in terms of its ancient Greek context. Praxis in Greek thought refers to the activity of free-born citizens, not slaves; further, it refers to activity done for its own sake and not as a means to an end, especially not an economic end. This is what Marx, and Marxists, mean by "true production" (as opposed to "false production").
I am very drawn to the idea of praxis in this sense. As far as goals go, developing a society in which praxis is possible to all sounds like a great idea. But I'm wondering if the idea of praxis can be entirely consistent with the previous points about thought's material basis; Eagleton describes communism as a system in which material needs are met so that true production becomes possible, but it's not as though we ever stop having material needs. Hunger renews itself.
6. Negative depictions of communism tend to show uniform citizens robbed of any individuality, like drones or the Borg, but according to Marx, utter homogeneity is capitalism's dream, not communism's. Equality--the cry of the French Revolution and of liberals* everywhere--is a bourgeois value. Capitalism views employees as equal because they are interchangeable, like the parts of a machine. Anyone can work at McDonald's. Money, or exchange value, realizes the ultimate dream of equality: all money is perfectly equal because it is perfectly interchangeable. Under communism, Eagleton says, we are able to self-actualize (see the first point) and engage in praxis (see the fourth point), and it is these two things that produce individuality. No two things, and no two people, are exchangeable; people are unique.
While I'm suspicious of the dichotomy presented here, I found I was forced to agree with the general point that capitalism is no defender of individuality--for example, I can imagine little more uniform than suburbia--but there also seems to be a bit of equivocation between equality as a political right (equality before the law, gender equality) and equality as interchangeability. It's not a lot of equivocation, but there's just enough for room to quarrel.
7. Eagleton notes that Adorno described Marx "as an enemy of utopia for the sake of its realization," which I thought was a neat phrase. It means, roughly, that in order to achieve the utopian society you're looking for, one must make realistic assessments both about the world as it is and what must be done to make the world into what it ought to be. Dreamers can't realize their dreams if they don't wake up. Which seems obvious, but it's sometimes good to make seemingly-obvious claims explicit. (We might also be able to say that Marx was an enemy of philosophy for the sake of philosophy's realization.)
While these ideas are Marx's, as elucidated by Eagleton, I think what makes most of them interesting is that we don't need to be Marxists, or even communists, to find them interesting and consider whether they might be true. Certainly, if true, all of these ideas would be very useful to non-Marxists as well as to Marxists.
*It might seem a surprise to readers, especially American ones, that I use the word "liberal" a bit dismissively, since I must look rather liberal to them. In Canada we have (at least) three major political parties: the Conservatives, the Liberals, and the New Democrats. This reflects the spectrum of respectable political positions, those being conservative, liberal, and socialist (which in Canada does not mean communism so much as capitalism with many socialist reforms). There's a history of use here that I don't want to get into, but in brief liberalism is seen by some of the academics I learned from or hang out with as a failed attempt to make conservatism into something nicer that, ultimately, found more hygienic ways of achieving conservatism's unethical ends. Alternately, it could just mean "not left enough." I'm perhaps not as hard on liberalism as this mini-definition would imply, but we could say that I'm skeptical of liberalism's abilities to achieve its better intentions. So my use of "liberal" here does not mark me as conservative but, rather, as socialist.